Cascade chaos and its dynamic characteristics

The dependence of sensitivity on initial conditions is the essence of chaos. And the randomness of chaos originates from the high sensitivity to initial values, which is measured by the Lyapunov exponents. It is found in this paper that the cascade of chaotic systems can considerably improve the Lyapunov exponents of cascade chaos and other dynamic properties. Therefore, in this paper, we study the cascade of chaotic systems and the influence on dynamic performances of the cascade chaos, and we present the definition and conditions of chaotic system cascade. It is proved in theory that the Lyapunov exponent of cascade chaos system is a sum of Lyapunov exponents of cascade subsystems. Appropriate cascade for chaotic systems can increase system parameters and expand parameter regions of chaos mapping and full mapping, thereby enhancing initial condition sensitivity of chaotic map and security of chaotic pseudo-random sequences. For logistic map, cubic map and tent map, the dynamic characteristics of logistic-logistic, logistic-cubic and logistic-tent cascade are investigated in detail, verifying the improvements on dynamic characteristics of cascade chaos systems. The proposed chaotic cascade system can be used to generate better pseudo-random sequences for initial condition sensitivity and security.